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A282197 a(n) is the smallest number d if the point (d,d) is shared by exactly n different Dyck paths in the main diagonal of the diagram of the symmetries of sigma described in A237593.

%I A282197 #29 Feb 10 2017 01:13:39
%S A282197 1,2,7,15,52,102,296,371,455,929,1853,2034,4517,4797,5829,6146,6948,
%T A282197 17577,19818,18915,60349,78369,113010,110185,91650,85171,311321,
%U A282197 123788,823049,128596,1650408,1136865,415355,906771,2897535
%N A282197 a(n) is the smallest number d if the point (d,d) is shared by exactly n different Dyck paths in the main diagonal of the diagram of the symmetries of sigma described in A237593.
%C A282197 This sequence is not monotone since a(19) = 19818 > 18915 = a(20).
%C A282197 Additional values smaller than 5000000 are a(37) = 1751785, a(38) = 1786732, a(39) = 1645139, a(41) = 1308771 and a(44) = 3329668.
%C A282197 Sequence A128605 of first occurrences of gaps between adjacent Dyck paths appears to be unrelated to this sequence.
%C A282197 First differs from A279286 (which is monotone) at a(19). - _Omar E. Pol_, Feb 08 2017
%C A282197 a(n) = d if the point (d,d) belongs to the first vertical-line-segment of exactly length n found in the main diagonal of the pyramid described in A245092 (starting from the top). The diagram of the symmetries of sigma is also the top view of the pyramid. - _Omar E. Pol_, Feb 09 2017
%e A282197 The four examples listed in A279286 are also examples for this sequences.
%e A282197 a(20) = 18915 is in the sequence since it is the first time that exactly 20 Dyck paths meet on the diagonal though a concurrence of exactly 19 paths on the diagonal happens only later at a(19) = 19818.
%t A282197 a240542[n_] := Sum[(-1)^(k+1)*Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
%t A282197 (* parameter cL must be sufficiently large for bound b *)
%t A282197 a282197[cL_, b_] := Module[{centers=Map[0&, Range[cL]], acc={1}, k=2, cPrev=1, cCur, len}, While[k<=b, cCur=a240542[k]; If[Last[acc]==cCur, AppendTo[acc,cCur], len=Length[acc]; If[centers[[len]]==0, centers[[len]]=cPrev]; acc={cCur}; cPrev=cCur]; k++]; centers]
%t A282197 a282197[50, 5000000] (* data *)
%t A282197 (* list processing implementation useful for "small" arguments only *)
%t A282197 a282197F[n_] := Map[Last, Sort[Normal[Map[First[First[#]]&, GroupBy[Split[Map[a240542, Range[n]]], Length[#]&]]]]]
%t A282197 a282197F[50000] (* computes a(1) .. a(20) *)
%Y A282197 Cf. A128605, A237593, A240542, A245092, A279286.
%K A282197 nonn,hard,more
%O A282197 1,2
%A A282197 _Hartmut F. W. Hoft_, Feb 08 2017